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In principle, the electronic transition energies/frequencies for a given species of atom can be calculated by solving the time-independent many-body fermionic Schrodinger equation for $n$ electrons in a central potential (or the Dirac equation, for the fine structure or in other situations where relativistic effects are important).

Alternatively, you could directly perform spectroscopy on samples of that atomic species, and experimentally determine which frequencies of light they absorb/emit.

For simple atoms like hydrogen (and perhaps alkali metals, although I'm less sure about that), the electronic transition frequencies can be determined extremely precisely from first principles. For more complicated atoms with multiple interacting valence electrons, you need to either numerically tackle the many-body Schrodinger/Dirac equation or resort to approximations like the Hartree-Fock approximation, and either approach limits the accuracy of your results.

I would guess - although I'm not sure - that for atoms with simple electronic configurations, purely theoretical calculations can beat the precision of purely experimental measurements, while the opposite is true for atoms with complicated configurations of many interacting electrons. Is this the case, and if so, for which species of atoms does each technique (currently) yield more precise transition frequencies?

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